Quasi-ideal embeddings and Rees matrix covers for regular semigroups
Let \(\mathcal C\) be a class of semigroups. We say that a semigroup \(S\) is a locally \(\mathcal C\)-semigroup if \(eSe \in {\mathcal C}\) for every idempotent \(e \in S\). A semigroup \(S\) is called a locally isomorphic image of a regular Rees matrix semigroup over \(\mathcal C\) if \(S\) is an image under a homomorphism from a regular Rees matrix semigroup over \(\mathcal C\) which is one-to-one on \(eSe\) for every idempotent \(e \in S\). A subsemigroup \(Q\) of \(S\) is said to be a quasi-ideal of \(S\) if \(QSQ \subseteq S\). Let \(\mathcal C\) be one of the following classes: all semilattices, all inverse semigroups, all \({\mathcal L}\)-unipotent semigroups, all orthodox semigroups, all unions of groups, or all \(E\)-solid semigroups. Then a locally \(\mathcal C\)-semigroup \(S\) is a locally isomorphic image of a regular Rees matrix semigroup over \(\mathcal C\) if and only if \(S\) can be embedded as a quasi- ideal into a locally \(\mathcal C\)-semigroup with greatest \(\mathcal J\)-class. Moreover, if \(S\) is a locally inverse semigroup, or a locally \(\mathcal L\)- unipotent semigroup, or a locally orthodox semigroup, or a locally \(E\)- solid semigroup then both conditions hold. If \(S\) is a locally testable semigroup, or a locally union of groups then both conditions hold if and only if the principal ideals of \(S\) form a semilattice under intersection.
scientific article; zbMATH DE number 130503
scientific article